Lemma 13 ([Deformation estimate, with its uniformities]). Use the preceding localization, with j sufficiently large and >0 sufficiently small that the minimizing-sequence centers have a common interior margin in C . Fix a bound L<∞ and …

We give the realization, Taylor justification, and actual mixed-derivative estimate separately. Realization of the first field. Let Z_1,…,Z_b be the global holomorphic spanning fields on Y . In the immersion G , let A be their column matrix along f . If r=_ Y , enumerate all r -row, r -column minors a_i of A . On K_ j, ' the sum of their squared moduli has a common positive lower bound. Their sup norms and Dirichlet norms are bounded at the fixed parameters: the minors are smooth holomorphic functions of X on this compact, and D(X) is bounded. Set σ_i=a_i ∑_k|a_k|^2 , b_ ik =σ_i∂σ_k- σ_k∂σ_i. These (0,1) coefficients have bounded L^2(Δ) norm. Solve ∂Q_ ik =b_ ik , skew-symmetrically, with bounded W^ 1,2 norm. Explicitly, if b_ ik =b_ ik,0 \,dζ , take Q_ ik =4∂_ζΔ_D^ -1 b_ ik,0 , with the convention Δ=4∂_ζ∂_ ζ . Here Δ_D^ -1 is the zero-Dirichlet inverse on the unit disc. The unit disc has smooth boundary, the principal coefficients are the identity, and all lower coefficients vanish. The scalar Dirichlet estimate therefore gives \|Δ_D^ -1 b_ ik,0 \|_ W^ 2,2 (Δ) ≤C\|b_ ik,0 \|_ L^2(Δ) , proving the asserted W^ 1,2 bound. Linearity preserves skew symmetry. Smooth input on the closed disc gives smooth output there . Here is the precise boundary estimate. For a smooth function q on the closed disc with boundary Fourier coefficients a_k , let h(re^ iθ )=∑_k a_kr^ |k| e^ ikθ be its harmonic extension. Integration by parts gives ∫_Δ|∇q|^2 =∫_Δ|∇h|^2+∫_Δ|∇(q-h)|^2 ≥2π∑_k|k||a_k|^2. If q_0(r) is the angular average, then for 1/2≤r≤1 |q_0(1)|^2≤2|q_0(r)|^2 +2(1-r)∫_r^1|q_0'(s)|^2\,ds. Integrating in r and applying Jensen bounds the constant coefficient |a_0|^2 by C\|q\|_ W^ 1,2 (Δ) ^2 . The nonconstant L^2 Fourier sum is bounded by its |k| -weighted sum. Consequently \|q|_ ∂Δ \|_ L^2 ^2+D(q|_ ∂Δ )^2 ≤C\|q\|_ W^ 1,2 (Δ) ^2. Applied entrywise to Q_ ik , this bounds its boundary D and L^2 norms. The operator constants are fixed-disc constants; the input norms can still depend on K'_ j, . The functions g_i=σ_i+∑_kQ_ ik a_k are holomorphic and ∑_i g_i a_i=1 : differentiating uses ∑_k a_kσ_k=1 and ∑_k a_k∂σ_k=0 , while skew symmetry cancels the quadratic term in the sum. A W^ 1,2 trace need not be bounded. For a large number P let α be the scalar outer function with positive center value and boundary modulus r_P=(1+|Q|^2/P)^ -1 . All norms and products involving Q in the following boundary estimates refer to its trace on the circle. The maps Q↦(1+|Q|^2/P) have Lipschitz constant C/√P , and their L^2 norms are at most C\|Q\|_2/√P . The maps Q↦r_PQ have bounded Lipschitz constant and sup norm at most C√P . The phase of α is the Hilbert transform of r_P . Thus \|α\|_∞≤1, D(α)≤C D(Q)/√P, D(αQ)≤C D(Q), \|αQ\|_∞≤C√P. In the third estimate the √P from the bounded product multiplies the P^ -1/2 bound for the phase. Moreover, ⟨1-|α|^2⟩≤C\|Q\|_2^2/P, -α(0)=⟨(1+|Q|^2/P)⟩⟶0. Given η>0 , choose P uniformly for the sequence so that these last two quantities are at most η and D(α)≤1 . The boundary estimates bound the sup norms and D norms of αg_i . These functions are holomorphic, so the maximum principle also bounds their interior sup norms at the fixed parameters; it is not applied to the nonholomorphic products αQ . For each minor a_i , Cramer's rule supplies a vector c_i , holomorphic in the disc, with Ac_i=a_iUl . Namely use the adjugate on its selected rows and columns and set the other coefficient entries to zero. The identity follows by expanding the augmented determinants: every (r+1) -minor of (A,Ul) is zero because this matrix still has rank r . This also treats selected minors that vanish identically. Consequently c=∑_iαg_i c_i has bounded sup norm and D and obeys Ac=αUl . Write c=(c_1,…,c_b) and compose the complex flows of Z_1,…,Z_b with times z h c_1,…,z h c_b , starting at f . For a sufficiently small common complex parameter z this gives a holomorphic family f_z , smooth up to the circle, whose first field is V=.∂_z f_z|_0=αhE_fl. The existence radius is uniform at the fixed parameters, since the initial images lie in K_ j, ' and all flow times are bounded. On the boundary the displacement is m times a uniformly bounded smooth function. Because m≤δ_i^a , decreasing this radius makes each new δ_i comparable to its old value, with constants between 1/2 and 3/2 . Thus the spatial boundary restrictions persist. The variable w and all times t are unchanged, so no uniform margin below the upper time boundary is needed. The center stays in C by its already established margin. Uniform Taylor remainders at the fixed parameters. We spell this out because smoothness of individual discs does not by itself imply the uniform expansion needed for an infimum. On the controlled interior compact the flow map and all its derivatives through any fixed finite order are bounded. Differentiating in the disc variable introduces one derivative of f , h , or c . Their Dirichlet norms are bounded; for f use local metric equivalence on K_ j, ' and. Hence the family and its first three real parameter derivatives have bounded W^ 1,2 norms, with bounded sup norms for their values. Differentiating the area integrands through order three yields only bounded coefficients times products of at most two such first disc derivatives. Cauchy--Schwarz supplies uniform integrable bounds. The same reasoning applies to the Gauss map, whose derivatives are bounded on the controlled compact. At the boundary write β=h/m . The boundary values of f_z can be written in ambient coordinates as a smooth function of X , c , β , and z m , on a fixed compact range. More precisely, for either cutoff, δ_i(f_z)=δ_i(f)+m R_i(z,X,c,β), R_i(0,·)=0, where the real parameter derivatives through order three of R_i have bounded sup and D norms at the fixed parameters, by . Since m/δ_i=δ_i^ a-1 δ_ 3-i ^ a is smooth on the closed boundary region, the function δ_i(f_z)-δ_i(f) =(1+(m/δ_i)R_i) and its parameter derivatives through order three have uniformly bounded D norms. For reciprocal walls, use 1 δ_i(f_z) -1 δ_i(f) =-(m)/(δ_i^2)\, (R_i)/(1+(m/δ_i)R_i). The prefactor is smooth and bounded when a≥2 . Therefore all nonzero parameter derivatives of the wall costs through order three are uniformly bounded, even when the unchanged wall cost is large. The remaining boundary integrands N,d_ 0,j have bounded derivatives on K_j×[0,T] . Finally the quadratic seminorm of the logarithmic wall has three bounded parameter derivatives by its Hilbert space pairing formula. These observations prove a uniform O(|z|^3) remainder after parameter-circle averaging of the entire functional minus the test. The constants in this paragraph are allowed to depend on the interior compact and on η . The actual mixed energy derivative uses boundary constants only. Set V=dG\,V=αhUl on the circle. Equations -- imply \|V\|_∞≤L, D(V)^2≤ C_j(L)(1+ E+D(l)^2). Here only \|α\|_∞≤1 and D(α)≤1 were used; the corona coefficients have disappeared. For clarity, consider any one of the closed (1,1) forms ω in the three spatial area terms of. It is a smooth closed form on Y after pullback. In ambient coordinates near the boundary write its intrinsic tensor as i b_ p q (X)\,dX^p∧d X^q , extending the coefficients smoothly from G( W_j) near G(K_j) as above. Deformed boundaries stay intrinsically in W_j for a smaller parameter radius. Closure is used intrinsically before this extension. Cartan's formula and Stokes give .∂_z∂_ z ∫_Δf_z^*ω|_0 =i∫_0^ 2π [, b_ p q (X)V^p∂_θ V^q; , +(∂_ r b_ p q )(X) V^pV^r\, ∂_^q]\,dθ. One can also verify the identity using local potentials: the mixed parameter derivative equals the boundary normal derivative of ω(V, V) , with the corresponding Stokes normalization. In, the holomorphic dependence of Gf_z eliminates a mixed second parameter field. The identity is global; it does not require partitioning ω into nonclosed forms. The coefficient b has bounded C^2 norm determined only by K_j . For B=\|V\|_∞ , D(b(X)V), ≤C_j(D(V)+B D(X)),; D((∂b)(X)VV) , ≤C_j(BD(V)+B^2D(X)). Pairing each angular derivative by bounds the absolute value of by C_j(L)(D(X)^2+D(V)^2) . This proves the required estimate for all spatial areas. The w area is unchanged. In particular for the Euclidean area the identity reduces, up to its fixed normalization, to D(V)^2 . It remains to check the logarithmic seminorm; its unbounded argument cannot be handled by an unweighted smooth-composition assertion. On the boundary put Z=αβUl , so that V=mZ . Then \|Z\|_∞≤L, D(Z)≤C_j(L)(1+√ E+D(l)). Regarding p=δ_1δ_2 as a smooth ambient function, the tensors m∂ p, =p^ a-1 ∂p,; m^2 p, =p^ 2a-1 p -p^ 2a-2 ∂p⊗∂p extend smoothly to the closed boundary region. For q_z= p(f_z) their contractions against Z and (Z, Z) are exactly ∂_zq_z|_0 and ∂_z∂_ z q_z|_0 , respectively. Consequently each has D bounded by C_j(L)(1+√ E+D(l)) . Write b=∂_zq_z|_0 and c=∂_z∂_ z q_z|_0 , and let B_D be the sesquilinear pairing B_D(u,v)=∑_k|k|u_kv_k . Since q_z is real, differentiating this quadratic seminorm gives exactly .∂_z∂_ z D(q_z)^2|_0 =2D(b)^2+2 B_D(q,c). Since D(q)^2≤ E , Cauchy--Schwarz proves |.∂_z∂_ z E(f_z,w)|_0| ≤C_j(L)(1+ E+D(l)^2). All constants in use boundary coefficients only, independently of the realization and Taylor constants. Testing the infimum and removing the auxiliary loss. Every deformed value of is at least μ_ j, , while the values at z=0 approach it. Average over |z|=r , subtract the value at zero, and divide by r^2 . First take the minimizing-sequence limit with r fixed, and then r0 . The uniform remainder just proved yields a nonnegative lower limit for the mixed derivative of cost minus test. Its nonpenalty terms are ⟨|α|^2 T_b(l, l)⟩ -|α(0)|^2h(0)^2φ_ E,c (l(0),l(0)). The center frames and their inverses are bounded independently of interior area by Lemma 10 ([Disc frames and Gauss area]). Thus the test term is bounded, as is the boundary symbol by. Removing α changes these terms by at most C_ j,L,φ η . Equation has a coefficient independent of η . Send η0 after the sequence and radius limits. This proves with the stated uniformities. In particular, none of the constants controlling the shrinking realization radius multiplies the final E error.